Puzzle for January 12, 2023 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
Subtract the left and right sides of eq.2 from the left and right sides of eq.4, respectively: D - A - (C - A) = B - F - (D - F) which becomes D - A - C + A = B - F - D + F which becomes D - C = B - D Add D to both sides of the above equation: D - C + D = B - D + D which becomes eq.4a) 2×D - C = B
Hint #2
In eq.5, replace B with 2×D - C (from eq.4a): 2×D - C - D = D - C - E which becomes D - C = D - C - E In the above equation, subtract D from both sides, and add C to both sides: D - C - D + C = D - C - E - D + C which makes 0 = -E which means 0 = E
Hint #3
In eq.3, replace E with 0: F - 0 = B - A which makes eq.3a) F = B - A
Hint #4
In eq.4, substitute (B - A) for F (from eq.3a): D - A = B - (B - A) which becomes D - A = B - B + A which becomes D - A = A Add A to both sides of the above equation: D - A + A = A + A which makes eq.4b) D = 2×A
Hint #5
Substitute 2×A for D in eq.2: C - A = 2×A - F Add A to both sides of the equation above: C - A + A = 2×A - F + A which becomes eq.2a) C = 3×A - F
Hint #6
Substitute 2×A for D, (3×A - F) for C (from eq.2a), and 0 for E in eq.6: 2×A - (3×A - F) = (3×A - F) - 0 - F which becomes 2×A - 3×A + F = 3×A - F - F which becomes -A + F = 3×A - 2×F Add A and 2×F to both sides of the above equation: -A + F + A + 2×F = 3×A - 2×F + A + 2×F which simplifies to 3×F = 4×A Divide both sides by 4: 3×F ÷ 4 = 4×A ÷ 4 which makes ¾×F = A
Hint #7
Substitute (¾×F) for A in eq.4b: D = 2×(¾×F) which makes D = 1½×F
Hint #8
Substitute (¾×F) for A in eq.2a: C = 3×(¾×F) - F which becomes C = 2¼×F - F which makes C = 1¼×F
Hint #9
Substitute ¾×F for A in eq.3a: F = B - ¾×F Add ¾×F to both sides of the equation above: F + ¾×F = B - ¾×F + ¾×F which makes 1¾×F = B
Solution
Substitute ¾×F for A, 1¾×F for B, 1¼×F for C, 1½×F for D, and 0 for E in eq.1: ¾×F + 1¾×F + 1¼×F + 1½×F + 0 + F = 25 which simplifies to 6¼×F = 25 Divide both sides of the above equation by 6¼: 6¼×F ÷ 6¼ = 25 ÷ 6¼ which means F = 4 making A = ¾×F = ¾ × 4 = 3 B = 1¾×F = 1¾ × 4 = 7 C = 1¼×F = 1¼ × 4 = 5 D = 1½×F = 1½ × 4 = 6 and ABCDEF = 375604